OK, but I encourage you to think about it and figure out why you can't challenge the math in the examples I gave earlier. It's not a big deal for this conversation, but it's important in the bigger picture because this distinction (or non-distinction depending on the perspective) is a big way and the reason politicians disguise taxpayer handouts through the tax code, with special deductions or tax credits handed out to favored special interests like candy.
One more example. Assume the community consists of only three families - yours, mine and Bob and Mary. We each pay $100 each year in taxes ($300 total) to fund our police department with a budget of $300. The next year, city council passes a tax credit of $100 for any family who adopts a child, and Bob and Mary do so and wipe out their tax bill. Collections drop by $100 to $200. Only two things CAN happen - either we cut the pay of our police by $100, OR, you and me now have to pay $133 in taxes, and Bob and Mary only $33 (tax of $133 less adoption credit of $100).
If city council had said, "we the city will directly pay $100 to any couple that adopts" we have the same result. City collects $300, $100 goes to Bob and Mary, leaving only $200 for the police OR taxes have to increase to $133 on each family ($400 total - police of $300 plus the $100 grant to Bob and Mary), but Bob and Mary get a $100 government check that cuts their net tax to $33.